İSTANBUL TECHNICAL UNIVERSITY INSTITUTE OF SCIENCE AND TECHNOLOGY

EFFECT OF TIME INTEGRATION SCHEMES ON THE LINEAR TRANSIENT ANALYSIS OF COMPOSITE PLATES

M.Sc.Thesis by

Kıvanç ŞENGÖZ, B.Sc.

Department: Mechanical Engineering Programme: Solid Mechanics

JUNE 2006

İSTANBUL TECHNICAL UNIVERSITY INSTITUTE OF SCIENCE AND TECHNOLOGY

EFFECT OF TIME INTEGRATION SCHEMES ON THE LINEAR TRANSIENT ANALYSIS OF COMPOSITE PLATES

M.Sc.Thesis by Kıvanç ŞENGÖZ, B.Sc.

(503021520)

Date of submission : 08 May 2006 Date of defence examination: 16 June 2006

Supervisor (Chairman): Assoc. Prof. Dr. Erol ŞENOCAK

Members of the Examining Committee: Prof. Dr. Mehmet DEMİRKOL

Prof. Dr. Zahit MECİTOĞLU

ACKNOWLEDGMENTS

I would like to thank my supervisor Assoc.Prof.Dr. Erol Şenocak for his help and patience throughout thesis. I would also like to thank all others who contributed to this thesis. In particular, I wish to thank Cem Celal Tulum, Hakan Tanrıöver, Gökmen Aksoy, Kadir Elitok for providing useful comments and help. Sincere thanks go to my mother and brother for their never ended supports. To the memory of my father, May 2006 Kıvanç ŞENGÖZ

ii

TABLE OF CONTENTS

ABBREVIATIONS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOL ÖZET SUMMARY

1. INTRODUCTION 1.1 Scope of Work

2. MECHANICS OF LAMINATED COMPOSITE PLATES 2.1 Engineering Constants of Orthotropic Materials 2.2 Constitutive Relations for Plane Stress 2.3 Coordinate Transformation For Laminates 2.4 Classical and First Order Theories of Laminated Composite Plates 2.5 Displacement , Strain and Constitutive Equations For The

Laminated Composite Plates

3. NAVIER SOLUTIONS FOR CROSS PLY LAMINATED COMPOSITE PLATES

3.1 General Remarks About The Navier Solution 3.2 Spatial Variation of The Solution For Classical and Shear

Deformable Plates 4. BASICS OF THE TRANSIENT DYNAMIC ANALYSIS 4.1 General Remarks About Transient Dynamic Problems 4.2 Governing Equations For Structural Dynamics 4.3 Direct Time Integration methods 4.3.1 Central Difference Scheme 4.3.2 Houbolt Scheme 4.3.3 Newmark Scheme 5. LINEAR TRANSIENT DYNAMIC APPLICATION ON CROSS PLY COMPOSITE PLATE 5.1 Effect of Time Integration Schemes On Cross Ply Composite Plate 5.2 Effect of Plate Theories On The Transient Response 5.3 Verification of Transient Code with Sample Literature Problems and geometric nonlinearity effect in plates.

6. CONCLUSIONS AND RECOMMENDATIONS

REFERENCES BIBLIOGRAPHY

IVV

VIVIII

IXX

1

2

3 4 5 6 6 7 9

13 13

15

19 1920 21 24 26 27

30

33 38 48

49

51 53

iii

ABBREVATIONS

CSPT : Classical Plate Theory SHDT : Shear Deformation Theory FEM : Finite Element Method FEA : Finite Element Analysis FSP : Finite Strip Method

iv

LIST OF TABLES

Table 5.1 Comparison of Matlab navier solution and Abaqus fem solution 33Table 5.2 Effect of Implicit constant average acceleration scheme on the

transient response of cross ply plate…………………………… 36Table 5.3 Effect of time integration with Explicit central difference…….. 38Table 5.4 Effect of time integration with Houbolt implicit scheme……… 38Table 5.5 Effect of classical plate theory on the response of two layer

cross ply plate………………………………………………….. 41Table 5.6 Effect of shear deformation theory on the response of two layer

cross ply plate………………………………………………….. 42Table 5.7 Effect of layers and shear deformation on the center transverse

deflection………………………………………………………. 42

v

LIST OF FIGURES

Figure 2.1 : Schematic of laminas and laminate in composite plate…………. 3Figure 2.2 : Kinematics of plate theories…………………………………….. 8Figure 2.3 : Force , Moment Resultants & Laminate Coordinate system …... 11Figure 4.1 : Errors due to direct numerical integration……………………… 25Figure 4.2 : Effect of time integration schemes on the period elongation……. 29Figure 4.3 : Numerical Damping Ratios For Different Schemes…………….. 29Figure 4.4 : Effect of different Newmark β parameters on period elongation.. 30Figure 5.1 : Unit step loading………………………………………………… 32Figure 5.2 : Code accuracy referenced to Reddy’s solution………………….. 33Figure 5.3 : Central Transverse displacement comparison of implicit and

explicit schemes with time step 1 microsecond…………………. 34Figure 5.4 : Central Transverse acceleration comparison of implicit and

explicit schemes with time step 1 microsecond…………………. 34Figure 5.5 : Transverse displacement comparison of implicit and explicit

schemes with time step 5 microsecond…………………………..

35Figure 5.6 : Central transverse acceleration comparison of implicit and

explicit schemes with time step 5 microsecond………………….

35Figure 5.7 : Effect of time steps of implicit trapezoidal method on transverse

displacement……………………………………………………... 36Figure 5.8 : Effect of time steps of implicit trapezoidal method on transverse

displacement…………………………………………………….. 37Figure 5.9 : Unstability behaviour of central difference method with time

step 6 microseconds……………………………………………... 37Figure 5.10 : Effect of Houbolt time integration on transient response……….. 39Figure 5.11 : Effect of numerical damping of Houbolt time integration scheme 39Figure 5.12 : Effect of numerical damping of Houbolt time integration scheme

at time step 10 microseconds…………………………………….

40Figure 5.13 : Center normal stress for classical and shear deformation theory 40Figure 5.14 : Center transverse displacement for classical and shear

deformation theory………………………………………………. 41Figure 5.15 : Uniformly distributed blast pressure - time history for isotropic

plate……………………………………………………………… 43Figure 5.16 : Blast Pressure versus time for sample problem of Chen ……….. 44Figure 5.17 : Response of isotropic plate to blast pressure central deflection… 45Figure 5.18 : Chen’s [20] linear and nonlinear displacmenet results with

finite element and finite strip method ………………………….. 45Figure 5.19 : Response of isotropic plate to blast pressure central in plane

stress…………………………………………………………….. 45Figure 5.20 : Chen’s [20] linear and nonlinear stress results with finite

element and finite strip method…………………………………. 46Figure 5.21 : Central transverse displacement response of one layer

orthotropic plate subject to uniform step loading……………….. 47

vi

Figure 5.22 : Chen’s linear and nonlinear displacement results with finite element and finite strip method for the response of one layer orthotropic plate ……………………………………………….. 48

Figure 5.23 : Central in plane stress of one layer orthotropic plate subject to uniform step loading……………………………………………. 48

Figure 5.24 : Chen’s [20] linear and nonlinear results with finite element and finite strip method for the response of one layer orthotropic plate to uniform step loading…………………………………… 49

vii

LIST OF SYMBOLS

a, b : side lengths of the plate A : Extensional stiffness B : Bending-Extensional coupling stiffness C : Damping coefficient D : Bending stiffness E : Young’s modulus F : Force h : plate thickness I : Inertia G : Shear modulus k : Shear coefficient K : Bending curvature Mij : Elements of Mass matrix Mi : Moment resultants N : Force resultants P : Load Q : Stiffness coefficients t : time u : displacement in the x direction v : displacement in the y direction V, ∆ : general displacements νij : poison ratio for the ij direction ρ : density ε : Strain σ : Stress

θ : Ply angle β , γ : Newmark Integration Parameters

ψ : Rotation ω : natural frequency w : displacement in the z direction z : distance from the midplane

viii

ZAMAN İNTEGRASYON YÖNTEMLERİNİN KOMPOZİT PLAKALARIN

DOĞRUSAL ZAMANA BAĞLI ANALİZLERİ ÜZERİNDEKİ ETKİLERİ

ÖZET

Bu çalışmada sırasıyla 0 ve 90 derecelik fiber yönlendirmesine sahip iki tabakalı elastik kompozit bir plakanın basit mesnetli sınır koşulları altında, zamana bağlı ve sinuzoidal dağılımlı yükleme altında, yer değiştirme uzayında analitik çözümü belirlenmiş ve zaman bağli analiz için gerekli adi diferansiyel denklem takımı elde edilmiştir. Çözüm için diferansiyel denklem takımı numerik olarak zaman boyutunda direk integrasyon teknikleri kullanarak integre edilmiş ve üç farklı integrasyon metodunun zamana bağlı davranış üzerindeki etkileri belirlenmiştir. Gerekli seri çözümü ve adi diferansiyel denklemin zaman boyunca integre edilmesi işlemleri Matlab yazılımı kullanarak programlanmış ve plaka merkezi üzerindeki düşey deplasman ve düzlem içi gerilmeler sunulmuştur. Ayrıca klasik ve birinci dereceden plaka teorileri de programlanarak sonuç üzerindeki etkileri incelenmiştir. Elde edilen sonuçlarda Newmark kapalı zaman integrasyon metodu olan trapez metod, zaman uzayındaki cevabın belirlenmesinde numerik stabilite ve hassasiyet açısından diğer metodlara göre daha iyi yaklaşımlar yapmıştır. Houbult metodu ve merkezi farklar yöntemi daha büyük zaman adımlarında istenilen çozüm hassasiyetinden sapmalar göstermiştir. Houbult metodu kısa sureli ani dinamik yükleme içeren problemlerde iyi sonuçlar üretemeyen bir kapalı zaman integrasyonu olarak gözlemlenmiştir. Açık direk integrasyon tekniği olan merkezi farklar yöntemi koşullu numerik stabilite nedeniyle belirli bir zaman adımı büyüklüğünü geçtikten sonra ( 5 mikrosaniye üzerinde ) sonsuza giden sonuçlar vermiş ve bu yönden problemli bir integrasyon tekniği olarak değerlendirilmiştir. Plaka kenar uzunluğu kalınlık oranının 5 olduğu bir durum için kayma etkisini gözönünde bulunduran plaka teorisi ile elde edilen deplasman sonuçları klasik teori ile bulunan sonuçlardan yüzde otuz farklı elde edilmiştir. Klasik teori kayma etkilerini gözönünde bulundurmadığı için deplasman ve gerilme cevabını gerçek değerinin altinda hesaplayabilmiş, plaka frekansını ise olması gereken cevabın üstünde hesaplamıştır.

ix

EFFECT OF TIME INTEGRATION SCHEMES ON THE LINEAR

TRANSIENT ANALYSIS OF COMPOSITE PLATE

SUMMARY

For cross ply composite plate with two layers, under appropriate boundary conditions and sinusoidal distribution of the transverse load with step loading , the exact form of the spatial variation of the solution is obtained and the problem is reduced to the solution of a system of ordinary equations in time, which are integrated numerically using direct integration methods. Effect of time integration schemes on linear elastic dynamic response of the composite plate is investigated considering implicit dynamic schemes average acceleration, Houbolt and explicit scheme central difference. Necessary code for the solution of linear transient analysis of composite plates was programmed in Matlab considering the procedure outlined above. Furthermore influence of classical and first order theories are addressed for transient dynamic response and included as different matlab codes. Numerical results for the central deflection and center in plane stress are presented for comparison with the aim of the code. The results clearly show that trapezoidal implicit method ( Newmark method with γ =0.5 and β 25.0= ) has better approximation in time space in terms of accuracy and stability. In Houbolt scheme and explicit central difference scheme deviation from the accuracy of the solutions were observed at larger time steps compared with trapezoidal method. Houbolt is inappropriate time integration technique for short duration dynamic problems. Although explicit scheme has good correlation at smaller time steps, it suffers instability with time steps larger than 5 microseconds. So conditional stability of this method is important problem at these time steps. In the considered length to thickness ( a/h=5) ratio, thirty percentage of difference was observed in displacement response when shear deformation theory is used in plate behaviour. Although this ratio is so severe classical plate theory under predict the displacement and stress response while over predict the frequency of the response.

x

1. INTRODUCTION

With the increased application of composites in high performance aircraft, studies

involving the assessment of the transient response of the laminated composite plate

are receiving attention. Composites plate are susceptible to low velocity ( 150 ft/sn)

impact as transient loads such as dropped tools, runway stones and tire blow out

debris. Relatively thick layered composite plates also have significant armor

applications and exposed to explosive and nuclear blast transient loading with

high load intensity in short durations. Also the need for more profound understanding

of the response of laminated composites arise from potential use of such materials in

jet engine fan and compressor blades which are vulnerable to impact of foreign

objects. The linear elastic transient response of isotropic plates has been investigated

by several researchers. Reissman and his colleagues [3] analysed a simply supported

rectangular isotropic plate subject to a suddenly applied load, uniformly distributed

load over a rectangular area. Exact solution was obtained using classical three

dimensional elasticity theory and classical and improved theories. Hinton and his

associates [4] presented transient finite element analysis of thick and thin isotropic

plates.

Layered composite plates exhibit general coupling between the inplane

displacements and the transverse displacements and shear rotations. Consequently,

the response of composite plates is sensitive to the lamination scheme. Also due to

low transverse shear deformation effects are more pronounced in composites than in

isotropic plates. Moon [13] has investigated the response of infinite laminated

plates subjected to the transverse impact load at the center of the plate. Chow [17]

has employed the laplace transform technique to investigate the dynamic response of

the orthotropic laminated plates. Sun and his colleagues [10] have employed the

classical method of separation of variables with the Mindlin - Goodman procedure

for treating time dependent boundary conditions and dynamic loadings. Anderson

and Madenci [7] presented analytical solution of finite geometry composite plates

under transient surface loading including material damping.

1

1.1 Scope of Work

It is clear that transient simulation of composite plates with the explicit finite element

codes has provided useful information to the researchers as a new trend. Ease of

implementation of the complex contact algorithms, nonlinear material properties and

damage models in explicit codes has accelerated their usage in last years. On the

other hand drawback of this time integration method is the stability problem due to

nature of the finite difference time integration scheme and sensitivity degree of

response to chosen time step more than implict methods have. Therefore in this

work nature of the time integration schemes trapezoidal method of Newmark - Beta,

Houbult and central difference on the transient response of composite plates is

investigated. Also stability, time step sensitivity, amplitude decay and period

elongation issues are addressed. This will enable researchers to decide on certain

attributes of time integration choice for transient response of composite plates.

During that response importance of plate theory used in the formulation for the

kinematics of the plate is another scope especially in the transverse deformation of

plates. At this point impact of classical and first order shear deformation theories on

transverse deflection and inplane stress is investigated to shed light on the

importance of theories for the kinematics of plate deformation.

2

2.MECHANICS OF THE LAMINATED COMPOSITE PLATES

Basic block of the laminated composites is a lamina or ply. It is the typical sheet of

composite material. Fibre- reinforced lamina consist of many fibers embedded in a

matrix material. Unidirectional fiber –reinforced lamina exhibit the highest strength

and modulus in the direction of the fibers but they have very low strength and

modulus in the direction transverse to fibers.

Figure 2.1 Laminas and laminate for composite plate [2]

A laminate is a collection of laminate stacked to achieve the desired stiffness and

thickness. The sequence of various orientations of a fiber reinforced composite

layer in a laminate is termed lamination scheme or stacking sequence. Unidirectional

fiber – reinforced laminate (all lamina have the same fiber orientation) will be very

strong along the direction of the fibers but weak in transverse direction. The

laminate will be weak in shear also. If laminate has layers with fibers oriented at

3

30 or 45 degrees, it can take shear loads. Laminated composite structures also have

disadvantages. Because of the material properties between layers, the shear stresses

produced between layers, especially at the edges of a laminate, may cause

delamination [2].

2.1 Engineering Constants of Orthotropic Materials

The constitutive equations for a material which has three orthogonal planes of

material property symmetry characterized by nine independent material properties.

Fourth order tensor with fully anisotropic tensor has 81 constants.

ijijklij E εσ = (2.1)

with strain symmetry arguments reduced 36 constants and including thermodynamic

considerations set up 21 constants and orthotropic symmetry argument result in 9

constants.

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

23

13

12

33

22

11

66

55

44

332313

232212

131211

23

13

12

33

22

11

εεεεεε

σσσσσσ

QQ

QQQQQQQQQQ

(2.2)

In this case the material stiffness coefficients Qij can be expressed in terms of

engineering constants E1 , E2 , E3 , G12 , G23 , G13 , v12 , v23 , v13 These constants are

measured using simple tests like uniaxial tension or pure shear [2].

2.2 Constitutive Relations for Plane Stress

A plane stress state is defined to be one in which all transverse stresses are

negligible. Most laminates are typically thin and experience a plane state of stress.

For lamina in the x1 , x2 plane , the transverse stress components are σ33, σ13, σ23 .

Although these stress components are in small comparison to σ11 , σ12 σ22 they can

induce failures because fiber reinforced composite laminates are weak in the

transverse direction ( because the strength providing fibers are in the transverse x1,

x2 plane). Thus the transverse stress components are not always neglected in the

4

laminate analyses. However whenever they are neglected in a laminate theory, the

constitutive equations must be modified to account for the fact that

σ33 =0, σ44=0 , σ55=0, and σ13 =0, σ13 =0, σ23 =0, (2.3)

So stress-strain relations take the form

(2.4)

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

3

2

1

66

2212

1211

3

2

1

0000

εεε

σσσ

QQQQQ

Qij called the plane stress-reduced stiffnesses are given by

Q11= 2112

1

1 vvE

−

Q12= 2112

122

1 vvvE

−

Q22= 2112

2

1 vvE

−

Q66= 12G

(2.5)

Note that reduced stiffness involve four independent material constants E1 ,E2 ,v12 ,

G12. When the normal stress is neglected but the transverse shear stresses are

included equation 2.3 should be appended with the constitutive relations.

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

5

4

55

44

5

4

00

εε

σσ

QQ

, Q44 = G23 Q55 = G13 (2.6)

2.3 Coordinate Transformations For Laminate

Since the laminate is made up of several orthotropic layers (laminas), with their

material axes oriented arbitrarily with respect to the laminate coordinates, the

constitutive equations of each layer must be transformed from lamina coordinates

x1,2,3 to the laminate coordinates x, y, z.

k

xy

yy

xx

kk

xy

yy

xx

QQQQQQQQQ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

γ

εε

σ

σσ

662616

262212

161211

(2.7)

5

k

xz

yzkk

yz

xz

QQQQ

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

γ

γσσ

5545

4544 (2.7)

Where k specifies the layer and Qij’s are elements of the transformation matrix

depend on ply angle of kth layer ( fiber orientation) .

Q 11 = Q11 cos4θ + 2 (Q12 + 2 Q66 ) sin2θcos2θ + Q22 sin4θ

Q 12 = ( Q11 + Q22 - 4Q66 ) sin2θ cos2θ + 2 (Q12 sin4θ cos4θ )

Q 22 = Q11 sin4θ + 2 (Q12 + 2 Q66 ) sin2θ cos2θ + Q22 cos4θ

Q 16 = ( Q11 - Q12 - 2Q66 ) sin2θ cos3θ + 2 (Q12 sin4θ cos4θ )

Q 26 = ( Q11 - Q12 - 2Q66 ) sin3θ cosθ + 2 ( Q12 - Q22 + 2Q66 )

Q 66 = ( Q11 + Q22 - 2Q12 -2Q66) sin2θ cos2θ + Q66 (sin4θ + cos4θ )

Q 44 = ( Q44 cos2θ + Q55 sin2θ ) , Q 45 = ( Q55 – Q44 ) cosθ sinθ

Q 55 = ( Q55 cos2θ + Q44 sin2θ )

Q 45 = ( Q55 – Q44 ) cosθ sinθ

Q 55 = ( Q55 cos2θ + Q44 sin2θ )

(2.8)

2.4 Classical and First Order Theories of Laminated Composite Plates

Classification of Structural Laminate Plate theories

(1) Equivalent single –layer theory ( ESL) ( 2-D)

a ) Classical laminate theory

b ) Shear deformation and higher order laminated theories. For example Reddy’s

third order theory [2].

(2) Three dimensional elasticity theory (3-D)

a ) Traditional 3-D elasticity formulations

b ) Layerwise theories

(3) Multiple model methods (2-D and 3-D)

The equivalent single layer (ESL) theories are derived from the 3-D elasticity

theory by making suitable assumptions concerning the kinematics of deformation or

the stress state through the thickness of the laminate. These assumptions allow the

6

reduction of the problem to 2-D problem. In the three dimensional elasticity theory

or in a layerwise theory, each layer is modelled as 3-D solid.

Composite laminates are formed by stacking layers of different composite materials

and fiber orientation. By construction, composite laminates have their planar

dimensions one or two orders of magnitude larger than their thickness. Often

laminates are used in applications that require axial and bending strengths. Therefore

composite laminates are treated as plate elements.

The simplest plate theory is the classical laminate plate theory (CPLT) which is an

extension of the Kirchhoff (classical) plate theory to the laminated composite

plates. It is based on the displacement field [2].

u1(x; y; z; t) = u (x; y; t) - z* ∂ w/ x∂ u2(x; y; z; t) = v (x; y; t) - z* ∂ w/ y∂ u3(x; y; z; t) = w (x; y; t)

(2.9)

(u,v,w ) are the displacements components along the ( x,y,z) coordinate directions.

The displacement field implies that straight lines normal to the xy plane before

deformation remain straight and normal to the midsurface after deformation. The

Kirchhoff assumption amounts to neglecting both transverse shear and transverse

normal effects, deformation is entirely due to bending and in plane stretching .

The next theory is the first order shear deformation theory (FSDT ) which is based on

the displacement field [2].

u1(x; y; z; t) = u (x; y; t) + z* ψ x (x; y; t)

u2(x; y; z; t) = v (x; y; t) + z* ψ y (x; y; t)

u3(x; y; z; t) = w (x; y; t)

(2.10)

here t is the time u1 , u2 , u3 are the displacements in x, y, z directions respectively

u, v , w are the associated midplane displacements and ψ x and ψ y are the slopes

in xz and yz planes due to bending only. The FSDT extends the kinematics of the

CLPT by including a gross transverse shear deformation in its kinematics

assumptions, the transverse shear strain is assumed to be constant with respect to the

7

thickness coordinate. Inclusion of this form shear deformation allows the normality

restriction of the classical laminate theory to be relaxed. The first order shear

deformation theory requires shear correction factors which are difficult to determine

for arbitrary laminated composite plate. The shear factor depends not only on the

lamination and geometric parameters, but also loading and boundary conditions.

In both CLPT and FSDT, the plane stress state assumption is used and plane-stress

reduced form of the constitutive is used. In both theories the inextensibility and

straightness of transverse normals can be removed. Such extensions lead to second

and higher order theories [2].

Figure 2.2 Kinematics of plate theories [2].

In addition to their inherent simplicity and low computational cost, the ESL models

often provide sufficiently accurate description of global response for thin to

moderately thick laminates such as gross deflections, critical buckling loads and

fundamental vibration frequencies and associated mode shapes. Of the ESL theories,

the FSDT with transverse extensibility appears to provide the best compromise of

solution accuracy, economy, and simplicity. However The ESL models have

limitations that prevent them from being used to solve the whole spectrum of

composite laminate problems. First accuracy of the global response predicted by

8

the ESL models deteriorates as the laminate becomes thicker. Second the ESL

models are often in capable of accurately describing the state of stress and strain at

the ply level near geometric and material discontinuities or near regions of intense

loading – the areas where accurate stresses are needed most.

2.5 DISPLACEMENT, STRAIN AND CONSTITUTIVE EQUATIONS FOR

THE LAMINATED COMPOSITE PLATES

For CLPT following Kirchhoff hypothesis holds. Straights lines perpendicular to the

midsurface (transverse normals) before deformation remain straight after

deformation. The transverse normals do not experience elongation (they are

inextensible). The transverse normals rotate such that they remain perpendicular to

the midsurface after deformation. The first two assumptions imply that the transverse

displacement is independent of the transverse coordinate and the transverse normal

strain εzz is zero. The third assumption results in zero transverse shear strains. In the

formulations layers are perfectly bonded together and each layer is of uniform

thickness. The strains associated with the displacement field can be computed using

either nonlinear- strain displacement relations or the linear strain displacement

relations. General form of nonlinear strains [2]

Exx = ⎟⎠⎞

⎜⎝⎛

∂∂

xu + 0.5*

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂ 222

xw

xv

xu

Eyy = ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂yv + 0.5*

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

222

yw

yv

yu

Ezz = ⎟⎠⎞

⎜⎝⎛

∂∂

zw + 0.5*

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂ 222

zw

zv

zu

Exy=0.5* ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

xv

yu +0.5* ⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

yw

xw

yv

xv

yu

xu

Exz=0.5* ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

xw

zu +0.5* ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎠⎞

⎜⎝⎛

∂∂

zw

xw

zv

xv

zu

xu

Eyz=0.5* ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

yw

zv +0.5* ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

zw

yw

zv

yv

zu

yu

(2.11)

9

If the components of the displacements gradients are of the order of ε then the small

strain assumptions implies that the terms of the order ε2 are negligible in the

strains. If the rotations ∂ w/ x∂ and ∂ w/ y∂ of transverse normals are moderate

( say 10o or 15o ) then ε2 terms of these terms are small but not negligible compared

to ε so they should be included in strain displacement equation. In this case small

strain with moderate rotations is special from of the geometric nonlinearity and this

form of strains named Von Karman strains and theory is the Von Karman plate

theory. In most simple form with linear strain and displacement assumption and If

we assume that total strain is made up of in plane stretching and bending K

strains also known as curvatures

0iε i

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

6

2

1

60

20

10

12

22

11

KKK

z

ε

ε

ε

εεε

(2.12)

Where

ε01 = ⎟

⎠⎞

⎜⎝⎛

∂∂

xu , ε0

2 = ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂yv , ε0

6 = ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂yv + ⎟

⎠⎞

⎜⎝⎛

∂∂

xu

1K = ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

− 2

2

xw , = 2K ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

− 2

2

yw , = 6K ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂−

xyw2

2

(2.13)

Stress strain relationship [2]

kkk

QQQQQQQQQ

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

12

22

11

662616

262212

161211

12

22

11

εεε

σσσ

(2.14)

Force and moment resultants [2]

dzNNN

m

k

zk

zk∑ ∫

=

+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

1

1

12

22

11

12

22

11

σσσ

=∑ ∫=

+m

k

zk

zk1

1

k

QQQQQQQQQ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

662616

262212

161211

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

6

2

1

60

20

10

KKK

z

ε

ε

ε

dz (2.15)

10

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

662616

262212

161211

12

22

11

AAAAAAAAA

NNN

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

60

20

10

ε

ε

ε

+ ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

662616

262212

161211

BBBBBBBBB

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

6

2

1

KKK

(2.16)

m is the number of layer of the plate for the equation 2.15 and 2.16.

Figure 2.3 Force, Moment Resultants.

dzzMMM

m zk

zk∑ ∫

+

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

1

1

12

22

11

12

22

11

σσσ

=∑ ∫+m zk

zk1

1

k

QQQQQQQQQ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

662616

262212

161211

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

6

2

1

60

20

10

KKK

z

ε

ε

ε

z dz (2.17)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

662616

262212

161211

12

22

11

BBBBBBBBB

MMM

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

60

20

10

ε

ε

ε

+ ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

662616

262212

161211

DDDDDDDDD

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

6

2

1

KKK

(2.18)

Aij is called extensional stiffnesses, Dij the bending stiffnesses and Bij the bending

extensional coupling stifnesses.

( Aij , Bij , Dij ) = ∑m

1

( ) ( )dzQijzz mz

z

m

m

*,,141

2∫ (2.19)

In the case of FSDT, the kirchoff hypothesis is relaxed by removing the third part,

transverse normals do not remain perpendicular to the midsurface after deformation.

This amounts including transverse shear strains in the theory. The inextensibility of

11

transverse normals requires that w not be a function of the thickness coordinate.

Assuming displacement field [2]

u1(x; y; z; t) = u (x; y; t) + z* ψ x (x; y; t) , xzu ψ=

∂∂

u2(x; y; z; t) = v (x; y; t) + z* ψ y (x; y; t) , yzv ψ=

∂∂

u3(x; y; z; t) = w (x; y; t)

(2.20)

Also note that 33ε is zero is due to inextensibility of transverse normal.

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

+

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

6

5

4

2

1

60

50

40

20

10

12

13

23

22

11

KKKKK

z

ε

ε

ε

ε

ε

εεεεε

(2.21)

ε05 = ⎟

⎠⎞

⎜⎝⎛

∂∂

xw + ψ x , ε0

4 = ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

yw + ψ y (2.22)

Also ε01 , ε0

2 , ε03 have the the same description with CLPT

1K = ⎟⎠⎞

⎜⎝⎛

∂∂

xxψ

, = 2K ⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂

yyψ

, = 6K ⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂y

xψ + ⎟

⎠⎞

⎜⎝⎛

∂∂

xxψ

4K = 0 , = 0 5K

(2.23)

And same form of Inplane force resultants and moment resultants are holds for

FSDT.

For i,j =1,2,6 Moment and Force resultants takes general form as shown below.

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

Kjj

DijBijBijAij

MN

i

i ε (2.24)

( Aij , Bij , Dij ) = ∑m

1

( ) ( )dzQijzz mz

z

m

m

*,,141

2∫ (2.25)

12

In addition to this, transverse force resultants Qi for i = 4,5

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡=

⎭⎬⎫

⎩⎨⎧

5

4

5545

4544

5

4

εε

AAAA

QQ

(2.26)

Aij = ∑ m

( ) ( )dzQijkjki mz

z

m

m

**41

∫ (2.27)

and k is the shear coefficient.

13

3. NAVIER SOLUTIONS FOR CROSS PLY LAMINATED COMPOSITE

PLATES

In this thesis, the navier solution method is used to determine the spatial variation of

the transient solution .Thus typical dependent variable is expanded as

a( x,y,t) = ( ) ( )yxtHnm

nm ,,

, φ∑ (3.1)

where Hmn are suitable functions of that satisfy the bundary conditions and Hmn are

coefficients to be determined such that a( x,y,t) satisfies its governing equation.The

choice of a seperable solution form as above implies that the general spatial

variation is independent of time and its amplitude vary with time.

3.1 General Remarks For The Navier Solution

In the Navier method the generalized displacements are expanded in double

trigonometric series in terms of the unknown parameters. The choice of functions in

the series is restricted to those which satisfy the boundary conditions of the problem.

Substitution of the displacements expansions into the governing equations should

result in unique, set of algebraic equations among the parameters of the expansion.

The Navier solutions can be developed for rectangular laminates with two sets of

boundary conditions. Even for these boundary conditions, not all laminates permit

the solution

Equation of motion for SHDT in the presence of applied transverse forces, q as

14

N1,x + N6,y = I0 * u,tt + I1 * ψ x ,tt N6,x + N2,y = I0* v,tt + I2 ψ x ,tt Q4,x + Q5,y = I0 * w,tt + q(x,y,t ) M1,x + M6,y – Q4 = I2 * ψx ,tt + I1 * u,tt M6,x + M2,y – Q5 = I2 * ψ y,tt + I1 * v,tt ∂ M1/ = Mix∂ 1,x , u/ = u2∂ 2t∂ ,tt

(3.2)

Where P, R and I are the normal, coupled normal-rotary and rotary inertia

coefficients.

(P, R , I ) = = ( ) dzzzh

h

ρ*,,12/

2/

2∫−

∑m

( ) ( )dzzz mz

z

m

m

ρ*,,141

2∫ (3.3)

Using the Constitutive equations 2.20, equations of the motions 3.2 could be

expressed in terms of the displacements as follows [1].

[ ]{ } { } [ ]⎭⎬⎫

⎩⎨⎧+=

..λλ MfL ,{ }=λ [u,v,w,ψ x ,ψ y ]T (3.4)

L11 = A11 d11 + 2 A16 d12 + A66 d22 + P d tt

L12 = ( A11 + A66 ) d12 + A16 d22 + A26 d22, L13 = 0 L14 = ( B11 d11+ 2B16 d12 + B66 d22 + Rdtt

L15 = ( B12 d11+ B66 ) d12 + B26 d22 = L24

L25 = 2B26 d12 + B22 d22 + B66 d11 + R d tt L22 = A22 d22 + 2 A26 d12 + A66 dtt + P d tt L33 = -A44 d11 - 2 A25 d12 – A55 d22 + P d tt L34 = A44 d1 - A55 d2 , L35 = -A45 d1 - A55 d2 L44 = D11 d11+ 2D16 d12 + D66 d22 – A44 + Idtt L45 = (D12 + D16 ) d12 + D16 d11 + D26 d22 Idtt - A45 L45 = 2 D16 d12 + D22 d22 + D66 dtt + Idtt - A55

( 3.5)

15

Where di = / d∂ ix∂ ij = / ( i = j =1,2 ) and d2∂ ji xx∂ tt = / 2∂ 2t∂ M11 = M22 = M33 = P , M44 = M55 = R Mij=0 for i j (i,j=1,2…,5) ≠ F3 =q and F1 =F2 =F4 = F5=0

(3.6)

3.2 Exact Form Of The Spatial Variation of The Solution

The boundary initial- value problem associated with the forced motions of the

layered anisotropic composite plate involves solving the equation 3.3 subjected to a

given set of boundary and initial conditions. It is not possible to construct exact

solutions to equation 3.4 when the plate is of arbitrary geometry , constructed

arbitrarily oriented layers and subject to an arbitrary loading and boundary

conditions. However an exact form of the spatial variation of the solution to

equation 3.4 can be developed for two different lamination schemes and associated

boundary conditions, when the plate is of rectangular geometry and subjected to

sinusoidally distributed with respect to x, and y arbitrary with respect to time

transverse loading.

Considering the following simply supported boundary conditions. x =0,a v = w = ψ y = 0 ; N1=0 M1=0 ; y= 0,b u = w =ψ x = 0 ; N6=0 M2=0 ; The following form of the solution satisfies the boundary conditions u = , v = ( ) ( )yxtU

nmnm ,1

,, φ∑ ( ) ( )yxtV

nmnm ,2

,, φ∑ , w = ( ) ( )yxtW

nmnm ,3

,, φ∑ ,

ψ x = ( ) ( )yxtXnm

nm ,1,

, φ∑ , ψ y = ( ) ( )yxtYnm

nm ,2,

, φ∑ , (3.7)

Where

φ1 = cos αx sin βy , φ2 = cos βx sin αy , φ3 = cos βy sin αx (3.8)

Substituting equation 3.7 into equation 3.4 we find that solution to equation to 3.4

exist when the transverse loading is of the form

16

q ( x,y,t) = ( ) ( )yxtQnm

nm ,3,

, φ∑ ,

Qmn (t) = ∫∫ba

tyxqab 00

),,(4 sin (mπx/a ) sin(nπy/b) (3.9)

and the lamination scheme is such that ( which correspond to cross play lamination

scheme )

A16 = A26 = A45 = B16 = B26 =D16 = D26 = 0 (3.10)

Under these conditions equation 3.3 becomes

[M] { } + [C] { ∆ } = {F} ..∆ (3.11)

Thus for a given α = mπ/a , β = nπ/b where a and b side lengths of the

rectangular plate where

{ ∆ } = { Umn , Vmn , Wmn , Xmn , Ymn }T { F} = { 0 , 0 , Qmn , 0 , 0 }T

(3.12)

The elements of symmetric stiffness coefficient matrix based on the first order

plate theory [1].

C11 = A11 α 2 + B66 β 2 , C12 = ( A12 + A66 ) α β

C13 =0

C14 = B11 α 2 + B66 β 2 , C15 = ( B12 + B66 ) αβ

C22 = A66 α 2 + A22 β 2

C23 =0 , C24 = C15 , C25 = B66 α 2 + B22 β 2

C33 = A44 α 2 + A55 β 2

C34 = α A44 , C35 = A55 β

C44 = D11 α 2 + D66 β 2 + A55

C45 = ( D12 + D66 ) α β , C55 = D66 α 2 + D11 β 2 + A55

(3.13)

17

And the elements of mass matrix

M11 = M22 = M33 = P M44 = M55 = R Mij = 0 for i j ≠

(3.14)

Elements of the force matrix

F3 = Qmn F1 = F2 = F4 = F5 = 0

(3.15)

The load coefficients for a sinusoidally distributed transverse load

q(x,y,t) = H(t) q0 sin (πx/a ) sin(πy/b) (3.16)

It is a one term solution ( Qmn = q0 H(t) and m = n =1 ) and therefore it is closed

form solution. For other types of loads, the Navier solution is a series solution, which

can be evaluated for sufficient number of terms in the series. For uniform load

terms in the series

Qmn = (16 q0 H(t) / π2mn ) for m, n odd (3.17)

Thus for a given α = mπ/a , β = nπ/b where a and b are the side lengths of the

rectangular plate. We have system of 5 differential equations to be solved in time.

So we need to integrate 5x5 matrix differential equation in time for the vector { ∆ }

of the generalized displacements.

Using the same methodology and considering CLPT equation of motion

N1,x + N6,y = I0* u,tt - I1 * w ,x,tt N6,x + N2,y = I0* v,tt + I1 w ,x,tt M1,xx + 2*M6,yx + M2,xx = q(x,y,t) + I0 * w ,tt - I2 *[( w,xx, tt) + ( w,yy, tt)] + I1*[(u,x,tt) + (v,y,tt)]

(3.18)

18

The elements of symmetric stiffness coefficient matrix based on classical plate

theory,

C11 = A11 α 2 + A66 β 2 , C12 = ( A12 + A66 ) α β

C13 = - B11α 3 – ( B12 + 2B66 ) αβ2

C22 = A66 α 2 + A22 β 2 , C23 = - ( B12 + 2B66 ) αβ2 – B22 β3

C33 = D11 α 4 + 2 ( D12 + 2 D66 ) α 2 β 2 + D22 β 4

(3.18)

and the elements of mass and force matrix

M11 = I 0

M22 = I 0 , M33 = I 0 + I 2 (α 2 + β 2 )

F3 = Qmn , F1 = F2 = F4 = F5 =0

(3.19)

This case we need to integrate 3x3 matrix of differential equation in time. The set of

algebraic differential equation for any fixed m, n can be solved with state-space or

modal analysis method. Both methods are algebraically complicated and require

determination of eigenvalues and eigenfunctions. Therefore solution of ordinary

differential equation in time will be computed by approximate numerical solution

considering the integration schemes for second order differential equations. Basically

in this numerical integration method, time derivatives are approximated using

difference approximations ( or truncated Taylor series ) and therefore solution is

obtained only for discrete times and not as a continuous function of time. There are

two major steps in the solution process for transient analysis. First assume spatial

variation of the displacements and reduce the governing partial differential equations

to set of ordinary differential equations in time and solve the ordinary differential

equation exactly if possible or numerically.

19

4. BASICS OF THE TRANSIENT DYNAMIC ANALYSIS

In many engineering applications of dynamic situations the inclusion of dynamic

terms in the analysis of structures is essential. The term dynamic implies that inertia

terms must be included in the equations of equilibrium. In static loading, time

required to increase the magnitude of the applied load is longer than the period of the

lowest vibration mode. However depend on the loading rate, if the time required to

increase the magnitude of the applied load from zero to its maximum value is less

than half the natural period of the structure, dynamic excitation occurs so inertia

effects become important. Examples where such inertia effects are important include

the impact loading of the structures due to collision or crash, blast or explosive

conditions in which case the loading is of high intensity and is applied for a short

period of time ( about microseconds ) and seismic action where the structural

loading takes the form of prescribed acceleration history [15].

4.1 General Remarks about Transient Dynamic Problems

Generally transient dynamic problems are classified depending on the effect of the

spectral characteristic of the excitation on the overall structural response, as wave

propagation problems and inertial problems.

Wave propagation problems: The wave propagation problems are those in which the

behaviour at the wave front is of engineering importance and in such cases it is the

intermediate and high frequency structural modes that dominate the response

throughout the time span of interest. Problems of that fall into this category are shock

response from conventional weapons such as explosive or blast loading and

problems in which waves effect such as focussing reflections and diffractions are

important [16].

Inertial problems: All dynamic problems which are not wave propagation type can be

considered as inertial and here the response is governed by relatively small number

20

of low frequency modes. Problems of this type are often also called structural

dynamic problems [15].

Frequencies and modes refer to the eigenspectrum of the linearized structural

dynamics equation. Because of nonlinearities this spectrum can be expected to vary

with the state. The qualifiers low and intermediate and high refer to frequencies

whose associated wavelengths are much smaller than the characteristic acoustic

wavelengths, respectively.

As a broad guideline wave propagation problems are best solved by explicit

integration techniques whereas implicit integration techniques are more effective

for inertial problems. However, the relative economy of both approaches is also

influenced by the topology of the finite element mesh and by the type of computer.

In the last twenty years, significant advances have been made in the development

and application of numerical methods to solution of dynamic transient problems. A

primary factor in this progress has been the parallel development of large high speed

digital computers providing the faster execution times required to make the

solution of large engineering problem a feasible proposition. This has resulted in the

development of many commercially available codes such as Ansys, Abaqus, Mark,

Adina.

4.2 Governing Equations for Structural Dynamics

To solve transient structural or continuum mechanics problems numerically, the

governing hyperbolic partial differential equations are first discretized in space .This

procedure is called a semidiscreatization. The semidiscretization will reduce the

problem to a system of ordinary differential equations in time, which in turn must be

integrated to complete the solution process.

In dynamic analysis, the governing semi discrete equations of motion are obtained by

considering the static equilibrium at time t which include effects of acceleration,

dependent inertia forces and velocity dependent damping forces in addition to the

externally applied time dependent load and internal forces due to initial stresses in

the system. In essence direct numerical integration is based on two ideas

[23].

21

Firstly, the dynamic equilibrium equations are satisfied at discrete time intervals ∆t

apart. This means that basically static equilibrium, which includes the effect of

inertia and damping forces are sought at discrete time points within the interval of

the solution.

Secondly, variation of displacements, velocities and accelerations within each time

interval ∆t is assumed. Different forms of these assumed variations give rise to

different direct integration schemes, each of them has different accuracy, stability

and cost.

The available direct procedures can be further subdivided into explicit and implicit

methods each of with distinct advantages and disadvantages. Each approach employs

difference equivalents to develop recurrence relations which may be used in a step by

step computation of the response. The critical parameter in the use of each of these

techniques is generally the largest value of the time step which can be used to

provide sufficiently accurate results, as this is directly related to the total cost of

satisfactory results.

4.3 Direct Time Integration Methods

In direct integration the governing ordinary second order differential equations in

time resulting from semi - discretization of the structural system ( for example by

means of finite element method) given by equation 3.5 for linear structural dynamic

response are integrated using a numerical step by step procedure. The term direct

means that prior to numerical integration process no transformation of the equations

into different form is carried out. The critical parameter in the use of each of these

technique is generally the largest value of the time step which can be used to provide

sufficiently accurate results and this is directly related to the total cost of

satisfactory analysis.

Explicit methods: This technique employs finite difference methods and is

particularly well suited for short duration dynamical problems or wave propagation

problems such as structures subject to blast or high velocity impact . In the explicit

method the solution at time t + ∆t is obtained by considering the equilibrium

conditions at time t and such integration schemes do not require factorization of the

22

effective stiffness matrix in the step by step solution. Hence the method requires no

storage of matrices if diagonal mass matrix is used. Computational cost per time step

is generally much less for explicit methods and less storage is required than

implicit methods. Operations for the explicit method are relatively few in number

and are independent of the finite element mesh band or front width. However,

explicit time integration schemes are only conditionally stable and generally require

small time steps to be employed to insure numerical instability. Here, the step size

restriction is often more severe than accuracy considerations require. This restriction

limits the effectiveness of the approach when used to study of dynamic problems of

moderate duration, e.g. earthquake response problems. The conditionally stable

algorithms require the time step size employed to be inversely proportional to the

highest frequency of the discrete system. The explicit methods allow the

displacements at the current time step t + ∆t, to be found in terms of the known

displacements, velocities, and accelerations, at previous time step t. This enables one

to evaluate the material physics which can be used to generate new internal stresses

for use in the solution at the next step. Also very complex material models can easily

incorporated using this approach.

Implicit methods: This approach has its strength in the capability of treating inertial

problems such as low speed impact and seismic problems and modelling of complex

structural geometries. In the implicit method the equations for the displacements at

the current time step involve the velocities and accelerations at the current step itself,

t + ∆t. Hence the determination of displacements at t+ ∆t involves the solution of

structural stiffness matrix at every time step. Many implicit methods are

unconditionally stable for linear analysis and maximum time step length that can be

employed is governed by the accuracy of solution and not by the stability of the

integration process. Although the implicit methods usually require considerably more

computational effort per time step than explicit methods, the time step may be much

larger since it is restricted in size only by accuracy requirements. The time step in

most explicit methods on the other hand, is restricted only by numerical stability

requirements which may result in a time step much smaller than needed for the

requisite accuracy, thus increasing the cost of the accuracy.

23

Stability aspects: The most significant drawback of the central difference method is

its conditional stability. If too large time step is used, exceeding stability limit of the

system ∆t > ∆tcr the solution quickly blows up and may reach the overflow limits on

the computer so the computation becomes numerically unstable. Time step is limited

for stability by

∆t < ∆tcr ∆tcr =2/ωmax (4.1)

for undamped system where ωmax is the maximum natural frequency of the

modelled structure considering

[K]-ω2 [M] = 0 (4.2)

Because of round of errors the practical limit on ∆t is roughly 25 % less than ∆tcr

In finite element, ∆tcr is the time required for an acoustic wave to traverse the

element with the least traversal time . For homogenous strain elements, this condition

can be rewritten in terms of the acoustic wave speed c and the element length l in the

form as ∆tcr < l /c. This equation corresponds to Courant- Freidrick -Lewy condition

that courant number r = c ∆t/l be less than unity.

The direct integration begin with some known initial conditions displacements V0

velocity , or acceleration at time t = 0. The integration scheme establishes an

approximate solution V

•

0V 0

••

V

1 , , at time t = ∆t and so on for t= 2∆tV•

1V••

1V 2 …..etc.

Each method employs a different assumed variation of displacement velocity and

acceleration with in the time interval. The errors involved in the direct integration

method may result from truncation, from the use of lower order difference

approximations to the derivatives of continuous variables, from numerical

instability, from amplification of errors in previous time steps into later time steps

and from computational round off.

24

Figure 4.1 Errors due to direct numerical integration [23]

These errors may introduce a shift in the period and amplitude of the responses

which are called period elongation and amplitude decay respectively. The numerical

stability of the direct integration method depends on the modal characteristics of the

system.

4.3.1 Central Difference Method

In principle, finite difference expression that approximates the acceleration and

velocity in terms of the displacements can be used to solve the equation of motion.

M + C + KV = P••

V•

V (t) (4.3)

In which the velocity and accelerations are expressed as [6].

nV•

= [ ]1121

−+ −∆ nn VV

t , (4.4)

[ ]112 21−+

••

+−∆

= nnnn VVVt

V (4.5)

Where subscripts n-1, n, n+1 represent the time at (n-1)∆t , n∆t , and (n+1) )∆t

respectively with n = 0,1,2…..

The displacement at time (n+1)∆t is obtained by considering equation motion at

time n∆t,

M••

V n + C + KVnV•

n = Pn(t) (4.6)

25

Substitute the relations for ••

V n and of equation 4.4 and 4.5 into equation 4.5

gives

nV•

K 11 ++ = nn PV (4.7)

Where

K = t

CdtM

∆+

22

1+nP = nn VtMKP )2( 2∆

−− - (t

Ct

M∆

−∆ 22 )Vn-1

(4.8)

Note that K is not included in K so there is no need to invert K in central difference

method. In these relations the equation for Vn+1 involves Vn and Vn-1. Therefore in

order to calculate the solution at the first time step, special starting procedure must

be used as shown in equation 4.9

V-1 = V0 + ∆t •••

∆+ 02

0 5.0 VtV (4.9)

Also can be obtained directly from equation 4.5 using V••

0V 0 , and P•

0V 0 .

Considering the single dof system K, C, M and P reduce the k, c, m, p. Rearranging

and setting c=0 and p=0,

Xn+1 = [A] Xn (4.10)

Where

Xn+1= ⎥⎦

⎤⎢⎣

⎡ +

n

n

VV 1

Xn = ⎥⎦

⎤⎢⎣

⎡

−1n

n

VV

(4.11)

A = , ⎥⎦

⎤⎢⎣

⎡ −∆−0112 22 tω

ω 2 = k/m (4.12)

Matrix A is called the amplification matrix. Stability and accuracy of an integration

algorithm depend on the eigenvalues of this amplification matrix and in order to have

a stable solution, the spectral radius, i.e. the maximum eigenvalue of matrix A , has

to be smaller than 1. So time step is limited for stability by ∆t < ∆tcr where

∆tcr =2/ωmax for undamped system where ωmax is the maximum natural frequency

26

of the modelled structure. In order to have a stable solution using the central

difference explicit integration algorithm, time step should be smaller than critical

time step. When the frequency gets higher, the allowable time step will decrease.

This conditionally stable character is main disadvantage in using explicit central

difference method.

4.3.2 Houbolt Method

Houbolt used the following finite difference equations for velocity and accelerations

at time t = (n+1) ∆t

=+

••

1nV [ ]2112 4521−−+ −+−

∆ nnnn VVVVt

(4.13)

1+

•

nV = [ ]2111 29181161

−−−+ −+−∆ nnnn VVVV

t (4.14)

These equations were obtained from consideration of a cubic curve that passes

through four successive ordinates. In order to obtain solutions at time (n+1) ∆t we

employ the equations of the motion at time (n+1) ∆t and using previous equations

for velocity and accelerations we have equation

K 11 ++ = nn PV (4.15)

to solve Vn+1 in equation 4.15 we can use equation 4.16 assume that (C = 0) no

physical damping

K = KdtM

+2

1+nP = nn VtMP )5( 21 ∆

++ - ( 2

4tM

∆)Vn-1 + ( 2t

M∆

)Vn-2 (4.16)

As with the central difference method, this formulation needs a special starting

procedure. Houbolt used the formulas for the derivatives at the third point of the four

successive points along the cubic curve. The formulas are given the following

values for V-1 and V-2

V-1 = 2V0 + -V••

∆ 02 Vt 1

V-2 = 6∆t + 6 -8V0

••

V••

∆ 02 Vt 1 + 9 V0

(4.17)

27

When the same amplification matrix which is 3x3 this case eigenvalues of that

matrix always less than 1 so automatically satisfied in every time step chosen.

However should be selected in order to achieve proper accuracy. Also note that t∆

K include K so inverse of K needed in every time step. This is the characteristic and

main disadvantage of implicit methods in terms of solution time and storage

requirements.

4.3.3 Newmark Method

In the Newmark method, two parameters λ and β are introduced to indicate how

much of the acceleration enters into the relations for velocity and displacement at the

end of the time interval. Adopted relations [23]

1+nV = ⎥⎦⎤

⎢⎣⎡ ∆+∆−+∆+ +

•••••

122 )())(5.0( nnnn VtVtVtV ββ

1+

•

nV = ⎥⎦⎤

⎢⎣⎡ ∆+∆−+ +

•••••

1)1( nnn VtVtV γγ

(4.18)

The first equation can be used to express in terms of . Then second

substitution of this relationship into second equation gives in terms of

then can be obtained by substituting these two relations into the equation of motion

at time (n+1) ∆t. The result is

1+

••

nV 1+

•

nV

1+nV•

1+nV

K 11 ++ = nn PV and assuming C=0

K = KdtM

+2β

P n+1 = nn Vt

P )1( 21 ∆++ β

+ (t∆β

1 ) nV•

+ nV••

⎟⎟⎠

⎞⎜⎜⎝

⎛−1

21β

(4.19)

Once Vn+1 is obtained the corresponding velocity and acceleration vectors can be

computed using,

1+

•

nV = ⎥⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ ∆−−∆−−

∆+∆−+

•••

+

•••

nnnnnn VtVtVVt

VtV 21 )5.0()(1)1( β

βγγ

=+

••

1nV ⎥⎦⎤

⎢⎣⎡ ∆−+∆+−

∆

•••

+ nnnn VtVtVVt

212 )5.0(1 β

β

(4.20)

28

The stability and accuracy of the Newmark method, which is controlled by two

parameters β and γ , can be evaluated by examining the eigenvalues associated

with the amplification matrix. Method is unconditionally stable for γ =0.5

and β 25.0≥ . When γ =0.5 and β 25.0= it is known as trapezoidal rule. In

trapezoidal rule there is no inherent numerical damping but it has period elongation

sensitive the time step chosen on the other hand there is considerable amount of

numerical damping for low modes in Houbolt schemes [22].

Figure 4.2 Effect of time integration schemes on the period elongation [21].

Figure 4.3 Time step versus period elongation [21].

29

Figure 4.4 Effect of time integration schemes with different Newmark β parameter

on the period elongation versus time step [21].

30

5 . LINEAR TRANSIENT ANALYSIS APPLICATIONS

For the linear transient analysis, Navier solution for the rectangular cross ply

composite plate with simply supported boundary conditions was programmed in

Matlab. Three different time integration scheme , central difference as explicit

scheme, Newmark integration with γ =0.5 and β 25.0= which corresponds to

average acceleration scheme and Houbolt scheme as implicit schemes were

programmed separately to investigate the effect of each scheme on the response.

Also classical plate theory and shear deformation theory was programmed separately

to observe the effect of shear deformation on the linear transient response. Results of

the Matlab code were also verified using the Abaqus commercial finite element code

results. Furthermore code was applied to some linear transient problems that were

taken from literature and results were compared.

In all numerical examples zero initial conditions for velocity and displacement are

assumed. All computations were checked in double precision workstation. The

following geometrical and material data for the analysis are used.

a=b=250 mm a/b=1 : a and b are side lengths of the plate.

h= 50mm : h is the thickness of the plate.

The material properties for composite plate are as follows.

E2 = 2.1 x 106 N/cm2 : young modulus in the direction transverse to fibers.

E1=25 x E2 : young modulus in the fiber direction.

G12=G23=G13=0.5 x E2 : Shear Modulus

v12 = 0.25 : poison ratio

ρ = 8 x 10-6 Nsec2/cm4 : density

31

Loading has sinusoidal distribution in spatial coordinates,

q = q0 Sin(x/a) Sin(y/b) q0 = 10 N/cm2 (5.1)

It is guaranteed that with this load intensity , behaviour of plate response is in linear

limits. So there is no considerable deformation compared to thickness of the plate.

Thus geometric nonlinearity effect could be excluded with in this load intensity and

linearity assumption is reliable. In time, step loading was used for 250

microseconds so general form of the loading is,

q(x,y,t) = H(t) q0 sin (πx/a ) sin(πy/b) (5.2)

where H(t) is unit value for 250 microseconds.

Figure 5.1 Unit step loading

In table one results with the time step 1 microseconds was presented comparing

Abaqus finite element results . In this analysis shear deformation theory was used in

two solution . In Abaqus S4R element type was chosen based on SHDT so including

shear effects during analysis. Same simple supported boundary conditions and

loading type were used in Abaqus and Matlab. Mesh was 20x20 element for

rectangular plate which has convergent results for this element number. Also shear

stiffnesses in Abaqus K11, K12, K22 was used in composite section. Values of these

stiffnesses are A44, A45, A55 which were computed with the Matlab code. Also three

integration points through the thickness were used as basis.

32

Table 5.1 Comparison of Matlab Navier solution and Abaqus fem solution for two layer cross-ply (0/90 ) square plate under suddenly applied step loading with implicit Newmark method with γ =0.5 and β 25.0= , timestep = 1 microsecond

Time (microsecond)

Center Deflection W (cm x 103)

Central Normal Stress σxx at z=h/2 ( N/cm2)

Shear stress σxz at midside ( N/cm2)

t Matlab Abaqus Matlab Abaqus Matlab Abaqus

10 0.0077 0.0076 4.036 3.9 0.69 0.67

20 0.0037 0.0367 28.21 27.44 2.26 2.19

40 0.1472 0.1474 113.8 110.2 5.89 5.72

60 0.2922 0.2925 226.9 220.4 12.37 12

80 0.412 0.4119 319.2 309.2 16.34 15.85

100 0.4605 0.4606 357 347 18.94 18.4

120 0.4178 0.4176 323.2 313.2 15.96 16.19

140 0.307 0.306 233.2 225.6 12.58 12.2

160 0.1574 0.1567 119.4 115.7 12.57 12.2

180 0.0417 0.0410 30.6 29.2 6.532 6.32

200 0.0014 0.0014 0.742 .702 0.56 0.54

As shown in table 5.1 Navier and fem solutions are in agreement each other. Small

differences in displacements are due to discretization of spacial coordinates in fem.

Also fem is using stresses that is calculated at gauss points then interpolated to

nodes.

0 0 . 5 1 1 . 5 20

0 . 5

1

1 . 5

2

2 . 5

3

3 . 5

4

4 . 5

x 1 0 -4

T im e (s e c o n d )

Cen

ter t

rans

vers

e di

spla

cem

ent

C o d e a c c u ra c y re fe ra n c e d t o R e d d y s o lu t io n

o lu t io no lu t io n

Reddy Solution tion

(cen

timet

er)

Figure 5.2 Code accuracy referenced to Reddy’s solu

33

Rm

e d d y sa t l sa b

iNavier Solu

2 . 5 3

x 1 0 -4

tion [1].

5.1 Effect of Time Integration Schemes On Cross Ply Composite Plate

In figure 5.3 and 5.4 central displacement and acceleration behaviour of the plate

was shown comparing trapezoidal method and central difference scheme as implicit

and explicit solutions. At the time step of 1 microsecond there is good correlation

between two solutions.

0 0 .5 1 1 .5 2 2 .5 3

x 1 0 -4

0

0 .5

1

1 .5

2

2 .5

3

3 .5

4

4 .5

5x 1 0

-4

T im e (s e c o n d )

Cen

ter t

rans

vers

e di

spla

cem

ent W

(cm

)

t im e s t e p = 1 m ic ro s e c o n d s

E x p lic i tS o lu t io nIm p lic it S o lu t io n

Figure 5.3 Central Transverse displacement comparison of implicit and explicit

schemes with time step 1 microsecond.

0 0 .5 1 1 . 5 2 2 .5 3

x 1 0 -4

-2 .5

-2

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

2

2 .5x 1 0 5

Tim e (s e c o n d )

Acc

eler

atio

n

t im e s te p = 1 m ic ro s e c o n d s

E x p lic itS o lu t io nIm p lic it S o lu t io n

34

Figure 5.4 Central transverse acceleration comparison of implicit and explicit schemes with time step 1 microsecond

On the other hand there is considerable deviation in period of the response in explicit

central difference scheme with the time step of 5 microsecond as shown in figure 5.5

and 5.6

0 0.5 1 1.5 2 2.5 3

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10-4

Time (second)

Cen

ter t

rans

vers

e di

spla

cem

ent W

(cm

)

t ime s tep= 5 microseconds

Explic itSolutionImplic it Solution

Figure 5.5 Transverse displacement comparison of implicit and explicit schemes

with time step 5 microsecond

35

0 0 .5 1 1 .5 2 2 .5 3

x 1 0 -4

-2 .5

-2

-1 .5

-1

-0 .5

0

0 .5

1

1 .5

2

2 .5x 1 0 5

Tim e (s e c o n d )

Acc

eler

atio

n

t im e s te p = 5 m ic ro s e c o n d s

E x p lic itS o lu t io nIm p lic it S o lu t io n

Figure 5.6 Central transverse acceleration comparison of implicit and explicit schemes with time step 5 microseconds

In trapezoidal scheme solution time step between 1 and 10 microsecond has no

appreciable effect on the accuracy of the solution as noted in table 5.2.

Table 5.2 Effect of Implicit trapezoidal (constant average acceleration) scheme on the transient response of cross ply plate.

time step (microsecond) 0.1 1 5 10 20 25 28 30 50

Max center deflection w*1000 (cm) 0.46182 0.46183 0.4616 0.4615 0.4613 0.46117 0.455 0.4398 0.4396

Stress σxx (N/cm2) 358.2 357.9 351.7 353.8 350 340 112 112 120

period of response(T) (microsecond ) 195.4 196 200 200 200 200 224 224 240

Effect of several time steps for the trapezoidal method was shown in figure 5.6 and

5.8. In explicit central difference method, solution instability occurs with the time

steps larger than 5 microsecond as shown in figure 5.9. Initiation of instability is at

96 microsecond when time step of 6 microsecond is used. With the increase of time

step, initiation of instability shifted to earlier time in transient response as noted in

table 5.3.

36

Figure 5.7 Effect of time steps of implicit trapezoidal method on transverse

displacement

0 1 2 3 4

x 10 -4

0

1

2

3

4

5

6x 10 -4

Tim e

Cen

ter D

ispl

acem

ent

t im es tep= 5e-5t im es tep= 1e-5t im es tep= 8e-6t im es tep= 1e-6

(cm

)

(second)

9.4 9.5 9.6 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4

x 10-5

4.56

4.57

4.58

4.59

4.6

4.61

4.62

4.63

4.64

x 10-4

Time

Cen

ter D

ispl

acem

ent

effect of different time steps on implicit time integration ( trapezoidal method)

timestep=5e-5timestep=4e-6timestep=1e-6

Figure 5.8 Effect of time steps of implicit trapezoidal method on transverse displacement

37

0 1 2 3

x 10-4

-1

-0.5

0

0.5

1

1.5x 107

Time (second)

Cen

ter t

rans

vers

e di

spla

cem

ent

t ime step = 6 microseconds

explic it method displacement

Figure 5.9 Unstability behaviour of central difference method with time step 6 microseconds

Table 5.3 Effect of time integration Explicit central difference method with Integration parameters α=0.5 β = 0 (conditionally stable )

time step (microsecond) 0.1 1 5 6 8 10

Max center deflection w*1000 (cm) 0.46182 0.46180 0.4610Unstable

after

96e-6

Unstable after

56e-6

Unstable after

40e-6

Normal Stress σxx (N/cm2) 358.2 357.3 355

period of response(T)

(microsecond) 195.6 198 210

Houbolt scheme is addressed the amplitude of the response accurately with the time

step 0.1 microsecond and 1 microsecond however there is considerable differences in

the period of response with in these time steps as noted in table 5.4 . Also the effect

of time step on the accuracy of the displacement and stress solutions after 3

microseconds is appreciable in this method.

38

Table 5. 4 Effect of time integration with houbolt implicit scheme

time step

(microsecond) 0.1 1 3 6 20

Maxcenter deflection

w*1000 ( cm) 0.46182 0.4617 0.4602 0.4559 0.4154

Normal Stress

σxx (N/cm2) 361 360 128 123 110

period of

response(T)

(microsecond )

170 174 216 228 320

Another issue observed in Houbolt method is the amplitude decay as shown in

figure 5.10 and 5.11. This amplitude decay is increasing with the larger time steps.

At 10 microseconds it is about the 4 percent of the first peak a shown in fig 5.12.

0 0.5 1 1.5 2 2.5 3

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

x 10-4

Time (seconds)

Cen

ter t

rans

vers

e di

spla

cem

ent W

(cm

)

time steps= 3 microseconds

Implicit Solution houbult scheme

Figure 5.10 Effect of Houbolt time integration on transient response

39

Figure 5.11 Effect of numerical damping of Houbolt time integration scheme

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

x 10-4

4

4.1

4.2

4.3

4.4

4.5

4.6

x 10-4

Time (microseconds)

Cen

ter t

rans

vers

e di

spla

cem

ent W

(cm

)

time step=3microseconds

houbult scheme

(second)

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10-4

Time (seconds)

Cen

ter t

rans

vers

e di

spla

cem

ent W

(cm

)

time step=10 microseconds

houbult scheme

Amplitude decay

Figure 5.12 Effect of numerical damping of Houbolt time integration scheme at time step 10 microseconds.

5.2 Effect of Plate Theories On The Transient Response

40

Investigation of the effect of plate theories clearly show that classical theory

underpredict the maximum displacement and stress as shown in figure 5.12 and 5.13

Figure 5.13 Center normal stress for classical and shear deformation theory

Figure 5.14 Center transverse displacement for classical and shear deformation theory

Also classical theory overpredicted frequency of the response when length

thickness ratio (a/h) is 5 as noted in table 5.5 and 5.6.

Table 5.5 Effect of classical plate theory on the response of two layer cross ply plate

41

time step (microsecond) 0.1 1 5 10 25

Max center deflection w*1000 (cm) 0.3162 0.3162 0.3160 0.3140 0.3051

In plane stress σxx (N/cm2) at

z=-h/2 -0.8725 -0.8716 -0.8711 -0.8704 -0.8356

period of response(T) (microsecond) 166 166 170 160 200

Table 5.6 Effect of shear theory on the response of two layer cross ply plate.

time step (microsecond) 0.1 1 5 10 25

Maxcenter deflection w*1000

(cm) 0.4618 0.4618 0.4616 0.4615 0.46117

In plane stress σxx (N/cm^2)

z=-h/2 -0.8909 -0.89 -0.8858 -0.8849 -0.8831

period of response

(microsecond) 195.4 196 200 200 200

It is also observed that shear deformation theory is less sensitive to time step

compared with the classical plate theory in this range. Considering the effect of

number of layers on transient response of the composite plate, considerable large

deflection was observed in two layer cross ply composite plate configurations

compared to single and multiple configurations as shown in table 5.7. Also ten to

twenty percent smaller frequency was addressed in two layer configuration

compared to other configurations.

42

Table 5.7 Effect of layers and shear deformation on the center transverse deflection and period of simply supported cross ply(0/90/.......) composite square plates for a/h=5.

LaminationScheme

(0/90/...)

1

2

3

Layers

4

5

6

7

8

Classical plate

theory

0.128

110

0.3153

170

0.128

110

0.1493

120

0.128

110

0.1336

110

0.128

110

0.1325

110

deflection*103 (cm)

period(microsecond)

Shear deformation

plate theory

0.3556

180

0.4605

200

0.3378

170

0.295

160

0.292

160

0.281

160

0.283

160

0.276

140

deflection*103 (cm)

period(microsecond

5.3 Verification of Transient Matlab Code with Sample Literature Problems

and Evaluation on The Geometric Nonlinearity Effect in Plates.

For the verification of the Navier solution based linear transient Matlab code, two

problems from the article of the J.Chen [20] were solved with the code. Also

geometric nonlinearity effect was issued.

First thin isotropic plate was considered with a side length a=b= 508 mm and

thickness of 3.4 mm. Material properties used in the analysis,

E= 206840 MPa, density =7900kg/m3 , poison ratio =0.3.

The plate is subject to the blast loading which has uniform instensity over the

whole plate at any instant of time, but this intensity varies with time as shown in

figure 5.15.

43

0 100 200 300 400 500 600 700 800-5

0

5

10

15

20

pressure (Mpa x100)

time ( miliseconds)

Blast loading

microseconds

Figure 5.15 Uniformly distributed blast pressure - time history for isotropic plate

Figure 5.16 Blast Pressure versus time for sample problem of Chen [20].

44

Time dependency of loading used by Chen is shown in figure 5.16. In Matlab this

uniform loading is applied as step loading within the interval of 10 microseconds.

This is the only difference regarding the original loading of Chen.

Figure 5.17 Response of isotropic plate to blast pressure central deflection

0 100 200 300 400 500 600 700 800-5

0

5

10

15

20

t(miliseconds)

Wc (mm)

Center transverse displacement of isotropic plate under blast loadingbased on linear transient analysis

Good correlation was observed for linear transverse displacement ( figure 5.17 )

compared with the linear finite strip solution of the Chen ( figure 5.18 ).

Figure 5.18 Chen’s [20] linear and nonlinear results with finite element and finite

strip method for the response of isotropic plate to uniform blast pressure central displacement

45

Also good correlation was obtained in central normal stress at upper side of the plate

as shown in figure 5.19 and 5.20.

0 100 200 300 400 500 600 700 800-200

-100

0

100

200

300

400

500

time ( miliseconds)

Gxx center (Mpa)

Center Inplane stress at z=h/2

Figure 5.19 Response of isotropic plate to blast pressure central in plane stress

Figure 5.20 Chen’s [20] linear and nonlinear results with FEM and FSM

It is also addressed due to large geometric transverse displacement about the 3th times thickness, geometric nonlinearity should be included for the first case

46

In the second case, one layer orthotropic plate with uniform step loading was

considered. Geometric dimensions for side lengths are 250 mm, thickness is 5mm

and intensity of the load is 0.1 MPa. Orthotropic material properties are as follows

E 1 = 525 000 MPa , E2 = 21 000 MPa

G12=G13 = G23 = 10500 MPa

ν12 =0.25, ρ=800 kg/m3 .

For uniform loading coefficients in the double trigonometric series expansion of

loads in the navier method,

Qmn = - 16 q0 / π2 m n ( m, n = 1,3,5 …….) , q0 : load intensity (4.14)

In this study upper limit of m and n are 20. So computations are including two

hundred terms for accuracy consideration.

As shown in figure 5.21 and 5.23, dimensionless displacement and stress response

of plate is in agreement with solution of Chen’s [20] in figures 5.12 and 5.24. In this

load intensity it is also observed that there is no considerable change in linear and

nonlinear geometric approaches. This result could be predicted from the maximum

dimensionless displacement value which is w/h = 0.43. So when displacement is

0.43 times the thickness, it could be accepted as in the linear range so small rotations

assumptions in bending still works and gives close results to the geometric nonlinear

approach.

For small deflections ( w ≤ 0.2 h ) linear plate theory gives sufficiently accurate

results. By increasing the magnitude of the deflections beyond the certain level

( w ≥ 0.3 h ) however the lateral deflections are accompanied by stretching the

middle surface. The membrane forces produced by such stretching can help

appreciably in carrying the lateral loads . If the plate, for instance, is permitted to

deform beyond half its thickness, its load-carrying capacity is already significantly

increased. When the magnitude of the maximum deflection reaches the order of the

plate thickness ( w ≈ h), membrane action predominates. Consequently, for such

problems, the use of an extended plate theory which accounts for the effects created

by large deflections is mandatory [24].

47

Figure 5.21 Central transverse displacement response of one layer orthotropic plate

subject to uniform step loading

0 20 40 60 80 100 120 140 160 180 200-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time ( miliseconds)

W(dimensionless)

Center transverse displacement for unifrom step loadingfor one layer orthotropic plate

x10-2

Figure 5.22 Chen’s [20] linear and nonlinear results with finite element and finite

strip method for the response of one layer orthotropic plate to uniform step loading with central displacement – time response.

48

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

time ( milisec)

Central Stress Gyy ( dimensionless)

Central Inplane stress Gyy at z=h/2

Figure 5.23 Central in plane stress of one layer orthotropic plate subject to uniform step loading

Figure 5.24 Chen’s [20] linear and n

strip method for the resp

step loading with centra

t ( milisecond )

onlinear results with finite element and finite

onse of one layer orthotropic plate to uniform

l stress –time at upper side of the plate.

49

7. CONCLUSIONS AND RECOMMENDATIONS

Transient analysis of composite plates has an important area of application in

aerospace and ballistic industry. Response of composite plates to drop, impact and

blast type of loadings is influential factor in design consideration. Thus in last years

several numerical solution techniques are developed and investigated to address this

type of response. Especially many finite element code exist in the commercial market

with implicit and explicit dynamic solvers. In this thesis, the Navier solution

technique was programmed with Matlab code to investigate the linear transient

response of cross ply composite plates. Effects of implicit and explicit time

integrations techniques were addressed. Also accuracy, stability and time step

sensitivity of these methods were outlined considering period elongation and

amplitude decay in displacement and in plane stress response. Effect of shear

deformation plate theory and number of layers on transient response of cross ply

composite plate is presented.

The results clearly show that trapezoidal implicit method ( Newmark method with

γ =0.5 and β 25.0= ) has better approximation in time space in terms of accuracy

and stability. It has no amplitude decay in response. Also this method gives

advantage of using larger time steps without sacrifice on the accuracy of the

solutions. In Houbolt scheme and explicit central difference scheme deviation from

the accuracy of the solutions were observed at larger time steps compared with

trapezoidal method. Although explicit scheme has good correlation at smaller time

steps, it suffers instability with time steps larger than 5 microseconds. So conditional

stability of this method is important problem at these time steps. This may result in

storage problems during the solution. Houbolt method is the most sensitive method

to time steps considered. It has shown serious inaccuracy in the considered time

step range compared with trapezoidal and central difference schemes. Also

considerable period elongation and numerical damping was observed in the transient

response for low mode of response. So Houbolt is inappropriate time integration

technique for short duration dynamic problems.

50

In the considered a/h ( a/h=5) ratio, thirty percentage of difference was observed in

displacement response when shear deformation theory is used in plate behaviour.

Although this ratio is so severe classical plate theory under predict the displacement

and stress response while over predict the frequency of the response. Difference in

results between two theory is lessening with the increase of a/h ratio. It is also

addressed that shear deformation theory is less sensitive time step in transient

dynamic problems compared with classical theory. As a result of the investigation on

the effect of number of layers, considerable large deflection was observed in two

layer cross ply composite plate configurations compared to single and multiple

configurations. Also ten to twenty percent increase in frequency was addressed in

two layer configuration compared to other configurations.

51

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53

BIBLIOGRAPHY

Kıvanç Şengöz was born in Nazilli at 1978. He enrolled Aeronautical and Aerospace

Department at Istanbul Technical University in 1997. He was graduated in 2001. He

led his professional career at FIGES A.Ş three years as a senior finite element

analyst. His master education at Solid Mechanics Program of the Mechanical

Engineering Department in ITU is going on.

54